Question

5.)

The average time to run the 5K fun run is 21 minutes and the standard deviation is 2.7 minutes. 45 runners are randomly selected to run the 5K fun run. Round all answers to 4 decimal places where possible and assume a normal distribution.

1. What is the distribution of XX? XX ~ N(,)
2. What is the distribution of ¯xx¯? ¯xx¯ ~ N(,)
3. What is the distribution of ∑x∑x? ∑x∑x ~ N(,)
4. If one randomly selected runner is timed, find the probability that this runner's time will be between 20.7962 and 21.3962 minutes.
5. For the 45 runners, find the probability that their average time is between 20.7962 and 21.3962 minutes.
6. Find the probability that the randomly selected 45 person team will have a total time more than 927.
7. For part e) and f), is the assumption of normal necessary? YesNo
8. The top 10% of all 45 person team relay races will compete in the championship round. These are the 10% lowest times. What is the longest total time that a relay team can have and still make it to the championship round? minutes

6.)

Each sweat shop worker at a computer factory can put together 4.1 computers per hour on average with a standard deviation of 0.9 computers. 17 workers are randomly selected to work the next shift at the factory. Round all answers to 4 decimal places where possible and assume a normal distribution.

1. What is the distribution of XX? XX ~ N(,)
2. What is the distribution of ¯xx¯? ¯xx¯ ~ N(,)
3. What is the distribution of ∑x∑x? ∑x∑x ~ N(,)
4. If one randomly selected worker is observed, find the probability that this worker will put together between 4.2 and 4.4 computers per hour.
5. For the 17 workers, find the probability that their average number of computers put together per hour is between 4.2 and 4.4.
6. Find the probability that a 17 person shift will put together between 66.3 and 68 computers per hour.
7. For part e) and f), is the assumption of normal necessary? YesNo
8. A sticker that says "Great Dedication" will be given to the groups of 17 workers who have the top 20% productivity. What is the least total number of computers produced by a group that receives a sticker? minutes (round to the nearest computer)

Answer 1

Mean = 21 minutes

Standard Deviation = 2.7 minutes

Sample size (n) = 45

a) Distribution of X follows normal distribution with parameters X~N(21,2.7)

b) As per central limit theorem, n > 30, So distribution of x̅ follows normal distribution with parameters x̅ ~ N(µ, σ/√n)

X~N(21,2.7/√45) = X~N(21,0.4025)

(c) The sampling distribution of sample sum is a normal distribution

mean = 45 x 21 = 945

standard deviation = √(45 * 2.7²) = 18.1121506177 18.1122

∑x follows normal distribution with parameters ∑x~N(945,18.1122)

(d) Probability that this runner's time will be between 20.7962 and 21.3962 minutes.

From Excel =NORMSDIST(-0.0755,TRUE)

From Excel =NORMSDIST(0.1467,TRUE)

=0.089